Onsager-Machlup Theory for Nonequilibrium Steady States and Fluctuation TheoremsTaniguchi, Tooru; Cohen, E.
doi: 10.1007/s10955-006-9252-2pmid: N/A
A generalization of the Onsager-Machlup theory from equilibrium to nonequilibrium steady states and its connection with recent fluctuation theorems are discussed for a dragged particle restricted by a harmonic potential in a heat reservoir. Using a functional integral approach, the probability functional for a path is expressed in terms of a Lagrangian function from which an entropy production rate and dissipation functions are introduced, and nonequilibrium thermodynamic relations like the energy conservation law and the second law of thermodynamics are derived. Using this Lagrangian function we establish two nonequilibrium detailed balance relations, which not only lead to a fluctuation theorem for work but also to one related to energy loss by friction. In addition, we carried out the functional integral for heat explicitly, leading to the extended fluctuation theorem for heat. We also present a simple argument for this extended fluctuation theorem in the long time limit.
A Nontrivial Scaling Limit for Multiscale Markov ChainsDeVille, R.; Vanden-Eijnden, Eric
doi: 10.1007/s10955-006-9237-1pmid: N/A
We consider Markov chains with fast and slow variables and show that in a suitable scaling limit, the dynamics becomes deterministic, yet is far away from the standard mean field approximation. This new limit is an instance of self-induced stochastic resonance which arises due to matching between a rare event timescale on the one hand and the natural timescale separation in the underlying problem on the other. Here it is illustrated on a model of a molecular motor, where it is shown to explain the regularity of the motor gait observed in some experiments.
The Boundary Structure of Zero-Temperature Driven Hard SpheresSotirov, Alexander
doi: 10.1007/s10955-006-9253-1pmid: N/A
We study the fundamental problem of two gas species whose molecules collide as hard spheres in the presence of a flat boundary and with dependence on only one space dimension. More specifically the steady linear problem considered is the one arising when the second gas dominates as a flow moving towards the boundary with constant microscopic velocity (and hence zero temperature). Theboundary condition adopted consists of prescribing the outgoing velocity distribution of the firstgas at the boundary. It is discovered that the presence of the boundary under general assumptions on the outgoing distribution ensures the convergence of a series of path integrals resulting in a convenient representation for the distribution of the velocities of the molecules returning at the boundary.
Nonequilibrium Gas and Generalized BilliardsDeryabin, Mikhail; Pustyl’nikov, Lev
doi: 10.1007/s10955-006-9250-4pmid: N/A
Generalized billiards describe nonequilibrium gas, consisting of finitely many particles, that move in a container, whose walls heat up or cool down. Generalized billiards can be considered both in the framework of the Newtonian mechanics and of the relativity theory. In the Newtonian case, a generalized billiard may possess an invariant measure; the Gibbs entropy with respect to this measure is constant. On the contrary, generalized relativistic billiards are always dissipative,and the Gibbs entropy with respect to the same measure grows under some natural conditions.
Lattice Boltzmann modeling with discontinuous collision components: Hydrodynamic and Advection-Diffusion EquationsGinzburg, Irina
doi: 10.1007/s10955-006-9234-4pmid: N/A
Irrespective of the nature of the modeled conservation laws, we establish first the microscopic interface continuity conditions for Lattice Boltzmann (LB) multiple-relaxation time, link-wise collision operators with discontinuous components (equilibrium functions and/or relaxation parameters). Effective macroscopic continuity conditions are derived for a planar implicit interface between two immiscible fluids, described by the simple two phase hydrodynamic model, and for an implicit interface boundary between two heterogeneous and anisotropic, variably saturated soils, described by Richard’s equation. Comparing the effective macroscopic conditions to the physical ones, we show that the range of the accessible parameters is restricted, e.g. a variation of fluid densities or a heterogeneity of the anisotropic soil properties. When the interface is explicitly tracked, the interface collision components are derived from the leading order continuity conditions. Among particular interface solutions, a harmonic mean value is found to be an exact LB solution, both for the interface kinematic viscosity and for the interface vertical hydraulic conductivity function. We construct simple problems with the explicit and implicit interfaces, matched exactly by the LB hydrodynamic and/or advection-diffusion schemes with the aid of special solutions for free collision parameters.
String Matching and 1d Lattice GasesMungan, Muhittin
doi: 10.1007/s10955-006-9247-zpmid: N/A
We calculate the probability distribution for the number of occurrences n of a given l letter word x inside a random string of k letters, whose letters have been generated by a known stationary stochastic process. Denoting by p(x) the probability of occurrence of the word, it is well-known that the distribution of occurrences in the asymptotic regime k → ∞ such that kp(x) ≫ 1 is Gaussian, while in the limit k→ ∞, and p(x) → 0 , such that kp(x) is finite, the distribution is Compound Poisson. It is also known that these limiting forms do not work well in the intermediate regime when kp(x) >rsim 1 and k is finite. We show that the problem of calculating the probability of occurrences is equivalent to determining the configurational partition function of a 1d lattice gas of interacting particles, with the probability distribution given by the n-particle terms of the grand-partition function and the number of particles corresponding to the number of occurrences on the string. Utilizing this equivalence, we obtain the probability distribution from the equation of state of the lattice gas. Our result reproduces rather well the behavior of the distribution in the asymptotic as well as the intermediate regimes. Within the lattice gas description, the asymptotic forms of the distribution naturally emerge as certain low density approximations. Thus our approach which is based on statistical mechanics, also provides an alternative to the usual statistics based treatments employing the central limit and Chen–Stein theorems.